You can install any kernel you want. There is no problem on using one or multiple Kernels at a time. But sometime you should consider the Kernel Mapping to your Ubuntu Version. For example following link lists out the mapping of different kernel to different version of OS:
Ubuntu to Mainline kernel version mapping
You can get a complete information of Kernel installing/removing from following link.
Kernel Mainline Builds
It helped me a lot to install a new kernel in Ubuntu 12.04.2. Now I can chose any kernel 3.5.0 or 3.8.0 from grub menu.
Following link illustrates how Ubuntu 12.04 LTS works actually. It illustates in very nice way, you would also like:
Ubuntu LTS Enablement Stack
Hope this short description helps you.
but I have no idea what the
3rd link i,e Ubuntu LTS Enablement Stack, if go through the figure, it see that Ubuntu 12.04.x LTS support and upgrade policy. Which states that the packages and kernels in Ubuntu 12.04 LTS will upgraded equivalent to non-LTS version. i,e in short:
12.04 ~ 12.04 (kernel 3.2.x)
12.04.2 ~ 12.10 (kernel 3.5.x)
12.04.3 ~ 13.04 (kernel 3.8.x)
12.04.4 ~ 13.10 (kernel 3.11.x or latest according to the time of release of 12.04.4. i,e why ?? in the figure)
Since 13.04 is called raring and 12.04.3 is using packages equivalent to 13.04. This is the reason 12.04.3 packages are called *-lts-raring-packages and the same you see in your repository. I hope your doubt would be clear. (Reply if still you've some confusion)
upgraded to 12.04.3. However, my Linux kernel has not been upgraded; it's 3.2.0-52-generic-pae.
This is because it is always good to stick with the kernel which best suits to your system. If your system is running fine with the kernel you are using, then their is no need to upgrade to a kernel. But if you've any problem (as you can see, I had one that I solved after upgrading kernel as well as a lot of xserver-xorg packages) then you can consider to upgrade your kernel.
Thus Ubuntu uses the same logic and doesn't upgrade your kernel. It just left for you to take change or say risk. :)
What is the best way to upgrade the kernel?
There is no any best way to upgrade a kernel. The way that worked for you is the best :)
I've mentioned the links to which you can use to install the latest kernel. But I recommend you to install the kernel more specific to your 12.04.3 i,e 3.8.x which are listing in your repository.
You can install using the command. Just copy and paste the it in terminal and execute:
sudo apt-get install linux-generic-lts-raring linux-headers-3.8.0-34 linux-headers-3.8.0-34-generic linux-headers-generic-lts-raring linux-image-3.8.0-34-generic linux-image-generic-lts-raring
or you can install it using Synaptic Package Manager. Type linux-generic-lts-raring and select it. It will automatically select all the latest -lts-packages to install. I'm posting what it shows to me:
Then just mark for installation and then Apply
You can also get help from this link.